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Finite geometries. Repr. of the 1968 ed. (English) Zbl 0865.51004

Classics in Mathematics. Berlin: Springer. xiv, 378 p. (1997).
The reviewer was honored to review this book when it was originally published (1968; Zbl 0159.50001). Everything we said in the original review still holds. We reiterate that this book contains a remarkable amount of information. Much of what has gone on since was greatly influenced by “the book.”
Perhaps it would be appropriate to indicate some of the more recent developments. Due both to limitations of the reviewer and to limitations of space, some things which should be listed will not be and none of them can be given more than a brief mention. Also, a complete list of references would extend the original set of 1200 by a few hundred more, so we will only give names. This will be very unfair to many people whose names should have been mentioned.
At the time the book was written, all known projective (or equivalently, all affine) planes of non-square order were translation planes or their duals. Most of the known planes of square order were obtainable from translation planes by a combination of dualities and derivations. The others were similarly related to the Hughes planes.
But the Hughes planes are, in a sense, coordinatized by quasifields and thus are closely related to translation planes. Since then, two sporadic types of planes of non-square order have been found. They are the Figueroa planes, which form a sort of cubic analogue of the Hughes planes and some planes with a single \((P,l)\) transitivity due a Coulter and Mathews.
These last were constructed by using the notion of planar functions introduced by the author and the reviewer.
There have been new constructions of translation planes mostly based on nests (A. Bruen, et al.) or on polar spaces, symplectic spaces, etc. (Kantor, Mason, Shult).
Most of the work in finite geometries (including the book) consists of generalizations of projective or affine planes or of what are called “combinational structures embedded in finite projective spaces” in the Math. Reviews classifications.
Most of what comes under “generalizations” may be included under Design Theory. One of several books on this subject: “Design Theory” by Th. Beth, D. Jungnickel and H. Lenz, Cambridge Univ. Press, N. Y. (1986; Zbl 0602.05001).
Under “structure” we may include nets, flocks (Thas), and generalized quadrangles (Payne). Norman Johnson has pointed out relations between these objects and translation planes. Partial geometries also are tied in with these subjects.
Much work has been done on \(k\)-arcs, ovals, and hyperovals, mostly by Italians. We mention Korchmarous (a Hungarian living in Italy) and Cherowitzo (an American of Italian descent), but there are many others.
The “buildings” of Tits have attracted much attention especially from people whose primary speciality is finite group.
Many of the methods developed for finite geometries does not really require finiteness. In many cases, the basic tool is a finite dimensional vector space over a field and the argument still works for a finite dimensional vector space over a division ring which is a finite dimensional extension of a subfield. Norman Johnson (and others) have worked on extending known results about finite geometries to the infinite case.

MSC:

51Exx Finite geometry and special incidence structures
51Axx Linear incidence geometry
51-02 Research exposition (monographs, survey articles) pertaining to geometry

Keywords:

finite geometry
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